On 17 juin, 20:17, quasi <qu...@null.set> wrote: > On Sun, 17 Jun 2012 02:20:49 -0700 (PDT), mluttgens > > <lutt...@gmail.com> wrote: > >83749105938571693046 = 157 + 83749105938571692889 > > = 277 + 83749105938571692769 > > = 283 + 83749105938571692763 > > = 487 + 83749105938571692559 > >etc... etc... etc... > > Notice that 157 is the _smallest_ prime that works. > > Thus, the following attempts all fail: > > 83749105938571693046 = 3 + 83749105938571693043 > 5 + 83749105938571693041 > 7 + 83749105938571693039 > 11 + 83749105938571693035 > 13 + 83749105938571693033 > 17 + 83749105938571693029 > ... > 151 + 83749105938571692895 > > So by trial and error, you find that 157 works, but you didn't > know that before you checked the other summand. So how did > you know in advance that some prime was going to work? > > To prove Goldbach's Conjecture, you have to show, for the > general even n > 4, not a just a particular case, that a pair > of odd primes exists whose sum is n. That you haven't done. > > Note: "mutatis mutandi" is not a valid principle of > mathematical reasoning. > > quasi > >
It was easy to show that the even number 26 is the sum of two primes:
26 = 3+23, 7+19, 13+13, 19+7 and 23+3.
Let's call those sums Goldbach's sums. The number 26 has thus 5 Goldbach's sums. Hence, the number 26 vindicates Goldbach's conjecture.
In order to falsify the conjecture, one should need to show that some even number can't be the sum of two primes.
Hereafter are the numbers n of Goldbach's sums for increasing even numbers N:
Clearly, when N increases, the number of Goldbach's sums also increases. Perhaps could somebody finds a formula linking N and n. The given example shows that n cannot decrease from some value of N and eventually becomes zero. Thus, it vindicates Goldbach's conjecture.